How does USB Disk prevents unauthorized access to data stored in the app using encryption?

The files stored in USB Disk are encrypted using Apple's "Data Protection" mechanism. See this page for details.

Files marked as protected in USB Disk are not readable (in an unencrypted form) while the device is locked, unless the file is open (like if a protected audio file is playing while the device is locked then it will keep on playing, but the next/previous protected audio file cannot be played). Files not marked as protected are also encrypted by the Data Protection mechanism but are readable after the first unlock of the device. All files are locked after a reboot, or if you force lock the device by pressing the power and volume buttons for a few seconds, or if your device is unused for 48 hours.

USB Disk Pro also lets you read and write encrypted zip files, using the "standard zip encryption" and AES methods (the later being the recommended method if you care about privacy). Files in the zip file are encrypted with a given key, on top of the Data Protection mechanism. Note that when you access files in an encrypted zip archive, they are temporarily decrypted. To make sure no decrypted data is available, close all viewers and choose "Clear cache" in the settings of the app.